cp's OEIS Frontend

A201552 Square array read by diagonals: T(n,k) = number of arrays of n integers in -k..k with sum equal to 0.

%I A201552 #49 Oct 16 2024 09:23:11
%S A201552 1,1,3,1,5,7,1,7,19,19,1,9,37,85,51,1,11,61,231,381,141,1,13,91,489,
%T A201552 1451,1751,393,1,15,127,891,3951,9331,8135,1107,1,17,169,1469,8801,
%U A201552 32661,60691,38165,3139,1,19,217,2255,17151,88913,273127,398567,180325,8953,1
%N A201552 Square array read by diagonals: T(n,k) = number of arrays of n integers in -k..k with sum equal to 0.
%C A201552 Equivalently, the number of compositions of n*(k + 1) into n parts with maximum part size 2*k+1. - _Andrew Howroyd_, Oct 14 2017
%H A201552 R. H. Hardin, <a href="/A201552/b201552.txt">Table of n, a(n) for n = 1..9999</a>
%H A201552 Michelle Rudolph-Lilith and Lyle E. Muller, <a href="https://doi.org/10.1016/j.disc.2015.04.001">On a link between Dirichlet kernels and central multinomial coefficients</a>, Discrete Mathematics 338.9 (2015): 1567-1572.
%H A201552 Wikipedia, <a href="http://en.wikipedia.org/wiki/Dirichlet_kernel">Dirichlet kernel</a>.
%F A201552 Empirical: T(n,k) = Sum_{i=0..floor(k*n/(2*k+1))} (-1)^i*binomial(n,i)* binomial((k+1)*n-(2*k+1)*i-1,n-1).
%F A201552 The above empirical formula is true and can be derived from the formula for the number of compositions with given number of parts and maximum part size. - _Andrew Howroyd_, Oct 14 2017
%F A201552 Empirical for rows:
%F A201552 T(1,k) = 1
%F A201552 T(2,k) = 2*k + 1
%F A201552 T(3,k) = 3*k^2 + 3*k + 1
%F A201552 T(4,k) = (16/3)*k^3 + 8*k^2 + (14/3)*k + 1
%F A201552 T(5,k) = (115/12)*k^4 + (115/6)*k^3 + (185/12)*k^2 + (35/6)*k + 1
%F A201552 T(6,k) = (88/5)*k^5 + 44*k^4 + 46*k^3 + 25*k^2 + (37/5)*k + 1
%F A201552 T(7,k) = (5887/180)*k^6 + (5887/60)*k^5 + (2275/18)*k^4 + (357/4)*k^3 + (6643/180)*k^2 + (259/30)*k + 1
%F A201552 T(m,k) = (1/Pi)*integral_{x=0..Pi} (sin((k+1/2)x)/sin(x/2))^m dx; for the proof see Dirichlet Kernel link; so f(m,n) = (1/Pi)*integral_{x=0..Pi} (Sum_{k=-n..n} exp(I*k*x))^m dx = sum(integral(exp(I(k_1+...+k_m).x),x=0..Pi)/Pi,{k_1,...,k_m=-n..n}) = sum(delta_0(k1+...+k_m),{k_1,...,k_m=-n..n}) = number of arrays of m integers in -n..n with sum zero. - _Yalcin Aktar_, Dec 03 2011
%F A201552 T(n, k) = the constant term in the expansion of (x^(-k) + ... + x^(-1) + 1 + x + ... + x^k)^n = the coefficient of x^(k*n) (i.e., the central coefficient) in the expansion of (1 + x + ... + x^(2*k))^n = the coefficient of x^(k*n) in the expansion of ( (1 - x^(2*k+1))/(1 - x) )^n. Expanding the binomials and collecting terms gives the empirical formula above. - _Peter Bala_, Oct 16 2024
%e A201552 Some solutions for n=7, k=3:
%e A201552 ..1...-2....1...-1....1...-3....0....0....1....2....3...-3....0....2....1....0
%e A201552 .-1....2...-2....2....2....2...-1....0....2....2...-2...-1...-2...-1....2...-1
%e A201552 .-3...-1....1...-3....2....1....0....1....3....0....2....0...-1....2...-2...-1
%e A201552 ..0....3....3....3...-2...-2....3....3...-3...-3....0...-1...-1...-1....0....3
%e A201552 ..2...-1...-1...-1...-3....0...-3...-2....1...-1...-1....1....1....0....3...-1
%e A201552 ..2...-1...-3....0....2....3....0....1...-2....1....1....1....3...-2...-3...-3
%e A201552 .-1....0....1....0...-2...-1....1...-3...-2...-1...-3....3....0....0...-1....3
%e A201552 Table starts:
%e A201552 .   1,      1,       1,        1,        1,         1,...
%e A201552 .   3,      5,       7,        9,       11,        13,...
%e A201552 .   7,     19,      37,       61,       91,       127,...
%e A201552 .  19,     85,     231,      489,      891,      1469,...
%e A201552 .  51,    381,    1451,     3951,     8801,     17151,...
%e A201552 . 141,   1751,    9331,    32661,    88913,    204763,...
%e A201552 . 393,   8135,   60691,   273127,   908755,   2473325,...
%e A201552 .1107,  38165,  398567,  2306025,  9377467,  30162301,...
%e A201552 .3139, 180325, 2636263, 19610233, 97464799, 370487485,...
%p A201552 seq(print(seq(add((-1)^i*binomial(n, i)*binomial((k+1)*n-(2*k+1)*i-1, n-1), i = 0..floor((1/2)*n)), k = 1..10)), n = 1..10); # _Peter Bala_, Oct 16 2024
%t A201552 comps[r_, m_, k_] := Sum[(-1)^i*Binomial[r - 1 - i*m, k - 1]*Binomial[k, i], {i, 0, Floor[(r - k)/m]}];  T[n_, k_] := comps[n*(k + 1), 2*k + 1, n]; Table[T[n - k + 1, k], {n, 1, 11}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Oct 31 2017, after _Andrew Howroyd_ *)
%o A201552 (PARI)
%o A201552 comps(r, m, k)=sum(i=0, floor((r-k)/m), (-1)^i*binomial(r-1-i*m, k-1)*binomial(k, i));
%o A201552 T(n,k) = comps(n*(k+1), 2*k+1, n); \\ _Andrew Howroyd_, Oct 14 2017
%Y A201552 Columns 1-10: A002426, A005191, A025012, A025014, A201549, A201550, A201551, A322538, A322539, A322540.
%Y A201552 Rows 3-10: A003215, A063496(n+1), A083669, A201553, A201554, A322535, A322536, A322537.
%Y A201552 Cf. A286928.
%K A201552 nonn,tabl,easy
%O A201552 1,3
%A A201552 _R. H. Hardin_, Dec 02 2011